0 5 10 15 20 25 30 35 40 45
1000 2000 3000 4000 5000
Time [sec]
Number of M2M devices
Average file upload time
Without relaying With relaying
Error bars indicate standard deviation
No error bars indicate zero standard deviation
Figure 4.23: Average FTP file upload time with and without RN
8,000 8,500 9,000 9,500 10,000 10,500 11,000
1000 2000 3000 4000 5000
[Kbits]
Number of M2M devices
Average traffic received in cell
Without relaying With relaying
Error bars indicate standard deviation
No error bars indicate zero standard deviation
Figure 4.24: Average traffic received in cell with and without RN
subscenarios. The future M2M traffic is expected to be based on a huge number of devices. The proposed scheme can be seen as a viable solution for capacity problems faced by the network operators.
The impact of the proposed RN based multiplexing scheme on the QoS provi-sioning to regular LTE-A traffic was also investigated. The FTP UEs were request-ing resources directly from the eNodeB and the M2M devices were served by the RN. The results showed that the proposed framework can significantly minimize the impact of M2M traffic on regular LTE-A traffic.
An analytical model of a system is developed with the primary goal of evaluating the system performance [Zak12]. Analytical modeling is required for the eval-uation and understanding of complex systems to avoid expensive field tests and exhaustive simulations. Analytical modeling helps in saving time and resources.
In many cases, it is not even possible to perform field experimentation. For in-stance, the performance evaluation of a metropolitan area network covering a city would require significant time, power and other resources. Simulation models are also usually very complicated with huge computational times for the evaluation of system performance. Understanding the complexities of a simulated system requires additional time. The motivation behind analytical modeling is the ap-proximation of the behavior of a system mathematically and reducing the time and effort required for processing.
In this thesis, an analytical model for the performance evaluation of the Relay Node (RN) data traffic aggregation and multiplexing scheme is presented. The model is based on the work of [JSC11] for multiplexing of small Common Part Sub-layer (CPS) packets into large ATM cells with the AAL2 layer multiplexer.
An ATM cell is made up of a 5 byte header and a 48 byte payload. Among the 48 bytes of payload, 1 byte is reserved for the start field. The rest of the 47 bytes are available for multiplexing the short CPS packets. The model features two queues, the multiplexing queue and the transmission queue. At the multiplexing queue, the short CPS packets wait for multiplexing until either the number of packets in the queue is sufficient to fill the ATM payload or a timer expires. In case, the timer comes into play, then the rest of the unused bits in the ATM payload are zero padded. The transmission queue is for the ATM cells with a maximum of r CPS packets in each ATM cell. The maximum number of packetsr in an ATM cell can be determined as
r= 47
l
(5.1) wherel is the fixed size of a CPS packet.
The comparison of results from simulation and analytical models provides val-idation to the simulation results if the results are in agreement with each other. It is common to have slight differences in the results primarily due to the reason that the analytical model is abstract and covers only the main features of a system. The simulation model is complex and it is very difficult to capture all the aspects of simulation in analytical model.
In order to further consolidate the notion that the simulation results are valid and acceptable, an abstract simulation model is also developed in this thesis, referred to as ‘simple’ simulation model. The purpose of the simple simulation model is to minimize the influence of complex simulation aspects and highlight the perfor-mance of the proposed scheme purely.
5.1 The Analytical Model
The approach in [JSC11] is analogous to the multiplexing scheme proposed in this thesis. The M2M data packets arrive at the Uu PDCP layer of the RN from various M2M devices. The arrival of a packet to the multiplexer at the RN PDCP layer sets a timer in such a way that the packet has to wait for a certain maximum duration of timeTmax for the arrival of more packets to achieve maximum multiplexing gain.
If the waiting time exceeds the timer expiry limitTmax, the multiplexing process is initialized and the multiplexed data is sent to the Un PDCP buffer of the RN. If all the Nmax PRBs are not required for transmission, the unused PRBs are allocated to other users requesting resources from the DeNB. In case, the maximum waiting timeTmax is not exceeded, but the accumulative size of the packets in the buffer ex-ceeds the TBS capacity (nmax−overhead), then the multiplexing starts even if the timer has not expired. The overhead from GTP, UDP, IP, PDCP, RLC, MAC and PHY layers of the Un interface of the RN have to be considered while determining the TBS capacity [Sem14]. The size of nmax depends on the TBS which depends on the channel conditions of the backhaul air interface. The maximum number of packetsr that can be multiplexed into a large IP packet can be expressed as
r=
nmax−overhead l
(5.2) wherel is the fixed size of the packet arriving at the RN Uu PDCP layer.
If there are k M2M packets in the PDCP buffer, then until the waiting time reachesTmax or the buffer size reachesnmax−overhead, the value ofk is such that 1≤ k< r. The arrival rate of the M2M data packets is considered to be exponen-tially distributed with a rate ofλ. The total waiting time ofkth packet arrival time
λ λ λ λ λ
p1 p2 p3 pr-2 pr-1
q1 q2 q3 qr-2 qr-1 qr
p0
stage 1 stage 2 stage 3 stager-1 stager
Figure 5.1: r-stage Coxian distribution
is Erlangian [JSC11]. The probability of the next arrival before multiplexing with k packets already in the buffer is pk, whereas the probability that the multiplexing process starts after the arrival ofkth packet isqk such that pk=1−qk. This process can be modeled as an r-stage Coxian process and is depicted in Figure 5.1. The probability of the next arrival with kpackets in the buffer can be determined using (5.3)
pk =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1, k=0
1− k−1
j∑=0
e−λTmax(λTmax)j
j! , 1≤k<r
0, k≥r
(5.3)
The probability of an arrival with 0 packets in the buffer, p0, is always 1 because it is not possible that the multiplexing would start without any packet in the buffer.
To multiplex packets, at least one packet in the buffer is required. Therefore, q0 is always 0. After the arrival of 1st packet, the system enters stage 1 of the r-stage coxian process shown in Figure 5.1 (i.e. there is a packet in the buffer). In this stage, it is possible that a 2nd packet arrives in the buffer before the timer expires with a probability p1. But it is also possible that there are no arrivals after the 1st one and the timer expires to start the multiplexing process, the probability of which isq1. Now if a 2nd packet arrives before the timer expiry, the process enters stage 2. In stage 2, there are two possibilities, either another packet arrives (p2) or the timer expires after 2nd arrival (q2). This can go on untilr arrivals, which is the maximum number of packets possible in the buffer. At stager, the only possibility
is that the multiplexing should start without waiting for any further arrivals. Thus, pr = 0 andqr = 1.
The Laplace-Stieltjes Transform (LST) of the Probability Distribution Function (PDF) of the r-stage coxian process can be expressed as
A(s) =
∑
rk=1
φk
λ s+λ
k
(5.4) whereφk is the path probability and can be expressed as
φk =qk
k−1
∏
j=0pj (5.5)
where p0= 1. Equation (5.5) implies thatφk = p1p2...pk−1qk.